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Chapter 1: Problem 33

Stars typically emit the red light of atomic hydrogen with wavelength \(656.3 \mathrm{nm}\) (called the \(\mathrm{H}_{\alpha}\) spectral line). Compute the wavelength of that light observed at Earth from stars receding directly from us with relative speed \(v=10^{-3} c, v=10^{-2} c,\) and \(v=10^{-1} c\)

Short Answer

Expert verified
The observed wavelengths for stars receding at speeds of \(v=10^{-3} c, v=10^{-2} c,\) and \(v=10^{-1} c\) are approximately \(656.9563\,\text{nm}, 662.863\,\text{nm},\) and \(721.93\,\text{nm}\) respectively.

Step by step solution

01

Understand the Doppler Effect for Light

The Doppler Effect describes the change in frequency (and thus wavelength) of a wave for an observer moving relative to the source of the wave. For light, when the source is moving away, the observed wavelength is longer (redshifted). The formula to calculate the observed wavelength due to the Doppler Effect is \[\lambda_{\text{obs}} = \lambda_{\text{source}} \left(1 + \frac{v}{c}\right)\] where \(\lambda_{\text{obs}}\) is the observed wavelength, \(\lambda_{\text{source}}\) is the emitted wavelength, \(v\) is the relative velocity of the source, and \(c\) is the speed of light.
02

Calculating the Observed Wavelength for \(v=10^{-3} c\)

Using the above formula, we can calculate the observed wavelength for a star receding at a speed of \(v=10^{-3} c\). Plug the values into the formula: \[\lambda_{\text{obs}} = 656.3\,\text{nm} \left(1 + \frac{10^{-3} c}{c}\right)\] Simplify to get the observed wavelength: \[\lambda_{\text{obs}} = 656.3\,\text{nm} \left(1 + 10^{-3}\right)\]
03

Calculating the Observed Wavelength for \(v=10^{-2} c\)

Following the same procedure as step 2 for a speed of \(v=10^{-2} c\): \[\lambda_{\text{obs}} = 656.3\,\text{nm} \left(1 + \frac{10^{-2} c}{c}\right)\] Simplify the expression: \[\lambda_{\text{obs}} = 656.3\,\text{nm} \left(1 + 10^{-2}\right)\]
04

Calculating the Observed Wavelength for \(v=10^{-1} c\)

Using the same equation for \(v=10^{-1} c\): \[\lambda_{\text{obs}} = 656.3\,\text{nm} \left(1 + \frac{10^{-1} c}{c}\right)\] Simplifying the equation yields: \[\lambda_{\text{obs}} = 656.3\,\text{nm} \left(1 + 10^{-1}\right)\]
05

Compute the Final Values for Observed Wavelengths

Calculate the exact values of the observed wavelengths for each velocity: \[\lambda_{\text{obs}}(v=10^{-3} c) = 656.3\,\text{nm} \times 1.001\] \[\lambda_{\text{obs}}(v=10^{-2} c) = 656.3\,\text{nm} \times 1.01\] \[\lambda_{\text{obs}}(v=10^{-1} c) = 656.3\,\text{nm} \times 1.1\] The products yield the observed wavelengths.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Redshift
The phenomenon of redshift is integral to the study of astronomy and the movement of celestial bodies. The term 'redshift' refers to the lengthening of the wavelength of light emitted by an object as it moves away from the observer. This effect is similar to the more commonly experienced Doppler Effect with sound, where a sound's pitch appears lower as its source moves away.

Imagine an ambulance siren's pitch dropping as it speeds away from you; similarly, light from a star shifts towards the red end of the spectrum, hence the term 'redshift'. In this context, 'red' doesn't necessarily mean the light is red鈥攊t indicates the change towards the longer wavelengths on the spectrum. The importance of redshift lies in its application in determining the relative speed and distance of distant galaxies and stars from Earth, playing a critical role in our understanding of the expanding universe.
Wavelength Calculation in Astronomy
Calculating the wavelength of light observed from celestial bodies allows astronomers to study various properties of the universe. Using the specific equation \[\lambda_{\text{obs}} = \lambda_{\text{source}} \left(1 + \frac{v}{c}\right)\]provided in the solution, we can see how the original emitted wavelength (\(\lambda_{\text{source}}\)) is adjusted based on relative velocity (\(v\)) and the speed of light (\(c\)).

To comprehend this further, consider \(\lambda_{\text{source}}\) as the untouched musical note played by an instrument鈥攖he 'true' pitch. When this instrument moves away from you, you perceive a pitch that varies from the original, this is the observed wavelength or \(\lambda_{\text{obs}}\). Through the provided formula, astronomers measure these slight variations in light's wavelength to determine how fast objects in space, such as stars and galaxies, are moving away from Earth.
The Significance of Spectral Lines
Spectral lines are unique fingerprints of elements in starlight, which appear as distinct lines within a star's spectrum. When an element, such as hydrogen, is heated to a high temperature, as it is in stars, it emits light at very specific wavelengths. These emissions create a pattern of lines鈥攕pectral lines鈥攖hat are characteristic to each element, akin to a barcode unique to each grocery item.

The \({\mathrm{H}}_{\alpha}\) spectral line at 656.3 nm, mentioned in the exercise, is a signature of hydrogen and is often used to detect changes in a star's movement and composition. When a star is receding from us, these spectral lines are redshifted, allowing astronomers to calculate the star's velocity relative to Earth. The precise analysis of these lines doesn't just reveal motion; it can also disclose the chemical composition, temperature, and pressure within the stars or galaxies being studied. This insight is crucial for piecing together the intricacies of our universe.

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Most popular questions from this chapter

Show that if a particle moves at an angle \(\theta\) with respect to the \(x\) axis with speed \(u\) in system \(S,\) it moves at an angle \(\theta^{\prime}\) with the \(x^{\prime}\) axis in \(S^{\prime}\) given by $$ \tan \theta^{\prime}=\frac{\sin \theta}{\gamma(\cos \theta-v / u)} $$

The proper mean lifetime of \(\pi\) mesons (pions) is \(2.6 \times 10^{-8}\) s. Suppose a beam of such particles has speed \(0.9 c\). ( \(a\) ) What would their mean life be as measured in the laboratory? (b) How far would they travel (on the average) before they decay? (c) What would your answer be to part \((b)\) if you neglected time dilation? \((d)\) What is the interval in spacetime between creation of a typical pion and its decay?

Two observers agree to test time dilation. They use identical clocks, and one observer in frame \(S^{\prime}\) moves with speed \(v=0.6 c\) relative to the other observer in frame \(S\). When their origins coincide, they start their clocks. They agree to send a signal when their clocks read 60 min and to send a confirmation signal when each receives the other's signal. ( \(a\) ) When does the observer in \(S\) receive the first signal from the observer in \(S^{\prime}\) ? (b) When does he receive the confirmation signal? (c) Make a table showing the times in \(S\) when the observer sent the first signal, received the first signal, and received the confirmation signal. How does this table compare with one constructed by the observer in \(S^{\prime} ?\)

Suppose that \(A^{\prime}, B^{\prime}\), and \(C^{\prime}\) are at rest in frame \(S^{\prime}\), which moves with respect to \(S\) at speed \(v\) in the \(+x\) direction. Let \(B^{\prime}\) be located exactly midway between \(A^{\prime}\) and \(C^{\prime} .\) At \(t^{\prime}=0,\) a light flash occurs at \(B^{\prime}\) and expands outward as a spherical wave. \((a)\) According to an observer in \(S^{\prime},\) do the wave fronts arrive at \(A^{\prime}\) and \(C^{\prime}\) simultaneously? (b) According to an observer in \(S,\) do the wave fronts arrive at \(A^{\prime}\) and \(C^{\prime}\) simultaneously? \((c)\) If you answered no to either \((a)\) or \((b),\) what is the difference in their arrival times and at which point did the front arrive first?

"Ether drag" was among the suggestions made to explain the null result of the Michelson-Morley experiment (see the More section). The phenomenon of stellar aberration refutes this proposal. Suppose Earth moves relative to the ether at velocity \(v\) and a light beam (e.g., from a star) approaches Earth at an angle \(\theta\) with respect to \(v\). ( \(a\) ) Show that the angle of approach in Earth's reference frame \(\theta^{\prime}\) is given by $$ \tan \theta^{\prime}=\frac{\sin \theta}{\cos \theta+v / c} $$ (b) \(\theta^{\prime}\) is the stellar aberration angle. If \(\theta=90^{\circ},\) by how much does \(\theta^{\prime}\) differ from \(90^{\circ} ?\)
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